7 Answers
7

The precise origin of the phrases "bull market" and "bear market" are obscure. The Oxford English Dictionary cites an 1891 use of the term "bull market". In French "bulle spéculative" refers to a speculative market bubble. The Online Etymology Dictionary relates the word "bull" to "inflate, swell", and dates its stock market connotation to 1714.[14] The fighting styles of both animals may have a major impact on the names. When a bull fights it swipes its horns up; when a bear fights it swipes down on its opponents with its paws.[15] When the market is going up, it is similar to a bull swiping up with its horns. When the market is going down it is similar to a bear swinging its paws down.

Perhaps because a bull is considered an aggressive animal, one that charges against its enemies, it has been connected to initiative and activity that marks an upward trend. On the other hand, bears spend a long time hybernating in winter, therefore being inactive, a trait that could have connected them to a downward trend in markets.

Some suggestions as to possible etymologies are made here citing various characteristics of bulls and bears; I don't believe a word of it myself, and I'd think a far better answer is to be found here; for what it's worth, I'd bet on the estimable Mr Quinion.

To summarise (for bear):-

A noble gentleman of this city, who has the honour of serving his country as major in the Train-bands, being at that general mart of stock jobbers called Jonathon’s, endeavouring to raise himself (as all men of honour ought) to the degree of colonel at least; it happened that he bought the Bear of another officer.

The Tatler, 7 July 1709. This tongue-in-cheek tale is saying that the
major, wanting to buy a promotion, speculated by selling some stock
short. When the transaction went wrong, the story goes on, the major
described his fellow officer as a bear-skin man, among other epithets,
and called him out, satisfaction being achieved through a fist-fight,
neither man being keen on firearms.

I just ran across a reference to "Bulls" and "Bears" in an anonymous open letter from Jonathan's—the "general mart of stock jobbers" cited in Brian Hooper's answer—to "the Treasury," published in London in (approximately) 1763:

You tell us a[t] your End of Town that we are to have the Liberty of cutting Logwood in the Bay of Honduras, and the free Navigation of the Ships laden therewith; ... that the French give up their Pretensions to the Captures made before the Declaration of War, and that we are to be paid the Expence we have been at for supporting the Prisoners here during the War, which were Objects we could never before obtain, and which we cannot compute by the opposite difference at less than two Millions and a half,—But pray will this Money circulate in the Alley? — Had it not been better that another Campaign should have taken place? Twenty Millions more had been raised for the ensuing Year ;—we might then have had the fingering of a little, for Jobs, Commissions, Contracts, Advance-money, Bulls, Bears, good News and bad News, Peace and War, Scrip. above and under Par, for one more Year. Our Harvest is at an End ; and if those who are Losers have a Right to complain, we certainly have the justest Cause to find Fault with the Preliminaries, rail at the M——r, and abuse the Plenipotentiary.

I can't tell what "Bulls" and "Bears" represent in this paragraph, but they seem very likely to have been set up in opposition to each other, much as "good News and Bad News," "Peace and War," "and Scrip. above and under Par" in the phrases that follow their appearance do. As far as I can tell, this is the earliest confirmed instance where "Bulls" and "Bears" in a financial context are mentioned together and with the strong suggestion of being opposites. That the opposition dates back to the 1760s is a surprise to me.

After doing research on line and also hearing it from a guide at the CA State Capitol, the terms Bull and Bear markets were apparently coined by the New York Times writer Horace Greely after watching an actual Bull and Bear fight in Toulumne Cty. CA in the 1850ies and refers to fighting styles of same. Back then it was supposedly a common form of entertainment (until the numbers of Grisly bears dwindled).
"Bear market" is when stocks are down like when Bears tried to pull the faces of the Bulls down to avoid the horns."
"Bull Market" refers to an upswing in the warket much like a Bull tries to fling the Bear after goring it.
Pretty appaling history we have, no?

One argument that has been put forward for Bull vs Bear is that bearskin traders would sell bearskins they did not own, and hope to make a profit by later purchasing skins from trappers at a lower price. This would be an early form of short selling where the delivery of the skins to the ultimate end use would occur later. Because a bearskin trader made money in a falling market, a falling market became known as a "bear market".

If this explanation is correct then the term "bull" market comes from the idea that people bull and bear were mortal enemies and if a bear market meant a falling market, then a bull market must mean the opposite.

I would not try to look for stories about animals. The bull was probably bill or French billet. A stock market where a great number of shares are offered for sale can be characterized by a word for paper such as bill or French billet.

A stock market situation where there are more people who want to buy shares is a buyer-market, and that word can easily change to bear-market. Bear and bull are terms typical of insider-language that raise the impression that only stock market specialists understand was is going on, for laymen those things are too difficult.

For those who want proofs I hereby state that this is my personal assumption and I have never up to now thought about how one might find evidence for this assumption. We won't find a declaration of someone who said I changed buyer-market to bear-market or bill to bull.

But I know that stock market quotations in German newspapers can have the additional information Geld (money) or Brief (letter). Geld means there were more buyers, Brief means there were more sellers of shares.

But perhaps it is possible to find similar expressions in English newapapers.

Edit
I see that my explanation is not correct. A bear market is a market with falling prices, where sellers are dominant. So bear should be "paper", i.e.
securities, which have paper form, are offered.
A bull market is a market with rising prices. So one might interpret bull as bill, i.e. bills of money are offered for buying securities.

I find it a bit strange that etymological dictionaries don't consider the possibility of word transformations ( words take on the shape of other words) when expressions are concerned that are devoid of any logical meaning. Instead one clings to the form of the words and wants to bend right the sense of the expressions with curious stories.